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For non-linear partial differential equations with initial-boundary value problems, based on the difference method and the optimization method with dynamic design variables, using unknown function values on discrete node points on time layer as design variables, the difference equations sets of all the discrete node points are constructed as stylized objective function. A layered accurate optimization algorithm about computing unknown function value on discrete node point is proposed. Universal computing program is designed, and practical examples are analyzed. Through comparing computation results with exact results, the effectiveness and the feasibility of proposed method are verified. The method can provide the condition for engineering application.
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Keywords:
- nonlinear partial differential equation /
- Initial-boundary value problem /
- optimization method with dynamic design variables /
- program design
[1] Gu C H, Li D Q, Shen W X 1994 Appliced Partial Differential Equations (Beijing: Higher Education Press) pp161–192 (in Chinese) [谷超豪,李大潜,沈玮熙 1994 应用偏微分方程(北京:高等教育出版社) 第161-192页]
[2] Liu S K, Liu S D 2000 Nolinear Equations in Physics (Beijing: Peking University Press) pp7–15 (in Chinese) [刘式适,刘式达 2000 物理学中的非线性方程 (北京:北京大学出版社)第7-15页]
[3] Taogetusang, Sirendaoerji 2010 Acta Phys. Sin. 59 5194 (in Chinese) [套格图桑,斯仁道尔吉 2010 物理学报 59 5194]
[4] Sun Z Z 2005 Numerical Method about Partial Differential Equations (Beijing: Science Press)pp73–80, 112,131–135 (in Chinese) [孙志忠 2005 偏微分方程数值解法 (北京:科学出版社)第73-80,112,131-135页]
[5] Lu J, Yan J R 2002 Acta Phys. Sin. 51 1428 (in Chinese) [卢竞,颜家壬 2002 物理学报 51 1428]
[6] Xie Y X, Tang J S 2004 Acta Phys. Sin. 53 2828 (in Chinese) [谢元喜,唐驾时 2004 物理学报 53 2828]
[7] Wang M L, Li X Z, Zhang J L 2008 Phys. Lett. A 372 417
[8] Taogetusang, Sirendaoerji 2009 Acta Phys. Sin. 58 5887 (in Chinese) [套格图桑,斯仁道尔吉 2009 物理学报 58 5887]
[9] Mo J Q 2011 Acta Phys.Sin. 60 020202(in Chinese) [莫嘉琪 2011 物理学报 60 020202]
[10] Cheng X P, Lin J, Yao J M 2009 Chin. Phys. B 18 391
[11] Li Y Z, Feng W G, Li K M, Lin C 2007 Chin. Phys. 16 2510
[12] Zheng L C, Feng Z F, Zhang X X 2007 Acta Phys. Sin. 56 1550 (in Chinese) [郑连存,冯志丰,张欣欣 2007 物理学报 56 1550]
[13] Guo Y C 2007 Introduction about Partial Differential Equations (Beijing:Tsinghua University Press) pp186–226 (in Chinese) [郭玉翠 2007 非线性偏微分方程引论(北京: 清华大学出版社)第186-226页]
[14] Ma W X, Gu X, Gao L 2009 Adv. Appl. Math. Mech. 1 573
[15] Chen L J, Ma C F 2010 Chin. Phys. B 19 010504
[16] Ma W X, Huang T W, Zhang Y 2010 Phys. Scr. 82 065003
[17] Hirota R 2004 The Direct Method in Soliton Theory (Cambridge: Cambridge University Press)
[18] Hietarinta J 2005 Phys. AUC 15 31
[19] W X Ma, Fan E G 2011 Comput. Math. Appl. 61 950
[20] Hou X L, Qian Y, Wu H T 2010 Acta Math. Eng. 27 663 (in Chinese) [侯祥林, 钱颖, 吴海涛 2010 工程数学学报 27 663]
[21] Hou X L, Liu T L, Zhai Z H 2011 Acta Phys. Sin. 60 090202 (in Chinese) [侯祥林,刘铁林,翟中海 2011 物理学报 60 090202]
[22] Polyanin A D, Zaitsev V F 2004 Handbook of Nonlinear Partial Differential Equations (Boca Raton, London, NewYork: Chapman & Hall/CRC) pp1–2
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[1] Gu C H, Li D Q, Shen W X 1994 Appliced Partial Differential Equations (Beijing: Higher Education Press) pp161–192 (in Chinese) [谷超豪,李大潜,沈玮熙 1994 应用偏微分方程(北京:高等教育出版社) 第161-192页]
[2] Liu S K, Liu S D 2000 Nolinear Equations in Physics (Beijing: Peking University Press) pp7–15 (in Chinese) [刘式适,刘式达 2000 物理学中的非线性方程 (北京:北京大学出版社)第7-15页]
[3] Taogetusang, Sirendaoerji 2010 Acta Phys. Sin. 59 5194 (in Chinese) [套格图桑,斯仁道尔吉 2010 物理学报 59 5194]
[4] Sun Z Z 2005 Numerical Method about Partial Differential Equations (Beijing: Science Press)pp73–80, 112,131–135 (in Chinese) [孙志忠 2005 偏微分方程数值解法 (北京:科学出版社)第73-80,112,131-135页]
[5] Lu J, Yan J R 2002 Acta Phys. Sin. 51 1428 (in Chinese) [卢竞,颜家壬 2002 物理学报 51 1428]
[6] Xie Y X, Tang J S 2004 Acta Phys. Sin. 53 2828 (in Chinese) [谢元喜,唐驾时 2004 物理学报 53 2828]
[7] Wang M L, Li X Z, Zhang J L 2008 Phys. Lett. A 372 417
[8] Taogetusang, Sirendaoerji 2009 Acta Phys. Sin. 58 5887 (in Chinese) [套格图桑,斯仁道尔吉 2009 物理学报 58 5887]
[9] Mo J Q 2011 Acta Phys.Sin. 60 020202(in Chinese) [莫嘉琪 2011 物理学报 60 020202]
[10] Cheng X P, Lin J, Yao J M 2009 Chin. Phys. B 18 391
[11] Li Y Z, Feng W G, Li K M, Lin C 2007 Chin. Phys. 16 2510
[12] Zheng L C, Feng Z F, Zhang X X 2007 Acta Phys. Sin. 56 1550 (in Chinese) [郑连存,冯志丰,张欣欣 2007 物理学报 56 1550]
[13] Guo Y C 2007 Introduction about Partial Differential Equations (Beijing:Tsinghua University Press) pp186–226 (in Chinese) [郭玉翠 2007 非线性偏微分方程引论(北京: 清华大学出版社)第186-226页]
[14] Ma W X, Gu X, Gao L 2009 Adv. Appl. Math. Mech. 1 573
[15] Chen L J, Ma C F 2010 Chin. Phys. B 19 010504
[16] Ma W X, Huang T W, Zhang Y 2010 Phys. Scr. 82 065003
[17] Hirota R 2004 The Direct Method in Soliton Theory (Cambridge: Cambridge University Press)
[18] Hietarinta J 2005 Phys. AUC 15 31
[19] W X Ma, Fan E G 2011 Comput. Math. Appl. 61 950
[20] Hou X L, Qian Y, Wu H T 2010 Acta Math. Eng. 27 663 (in Chinese) [侯祥林, 钱颖, 吴海涛 2010 工程数学学报 27 663]
[21] Hou X L, Liu T L, Zhai Z H 2011 Acta Phys. Sin. 60 090202 (in Chinese) [侯祥林,刘铁林,翟中海 2011 物理学报 60 090202]
[22] Polyanin A D, Zaitsev V F 2004 Handbook of Nonlinear Partial Differential Equations (Boca Raton, London, NewYork: Chapman & Hall/CRC) pp1–2
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